优化算法——拟牛顿法之L-BFGS算法

一、BFGS算法

在“优化算法——拟牛顿法之BFGS算法”中，我们得到了BFGS算法的校正公式：

利用Sherman-Morrison公式可对上式进行变换，得到

令，则得到：

二、BGFS算法存在的问题

在BFGS算法中。每次都要存储近似Hesse矩阵

B_k^{-1}" title="B_k^{-1}" alt="" />，在高维数据时，存储浪费非常多的存储空间，而在实际的运算过程中。我们须要的是搜索方向。因此出现了L-BFGS算法。是对BFGS算法的一种改进算法。

在L-BFGS算法中。仅仅保存近期的次迭代信息。以减少数据的存储空间。

三、L-BFGS算法思路

令

\rho&space;_k=\frac{1}{y_k^Ts_k}" title="\rho _k=\frac{1}{y_k^Ts_k}" alt="" />。

V_k=I-\frac{y_ks_k^T}{y_k^Ts_k}" title="V_k=I-\frac{y_ks_k^T}{y_k^Ts_k}" alt="" />，则BFGS算法中的

H_{k+1}" title="H_{k+1}" alt="" />能够表示为：

若在初始时，假定初始的矩阵

H_0=I" title="H_0=I" alt="" />，则我们能够得到：

H_{1}=V_0^TH_0V_0+\rho&space;_0s_0s_0^T" title="H_{1}=V_0^TH_0V_0+\rho _0s_0s_0^T" alt="" />

\begin{align*}&space;H_2&space;&=&space;V_1^TH_1V_1+\rho&space;_1s_1s_1^T\\&space;&=&space;V_1^T\left&space;(&space;V_0^TH_0V_0+\rho&space;_0s_0s_0^T&space;\right&space;)V_1+\rho&space;_1s_1s_1^T\\&space;&=&space;V_1^TV_0^TH_0V_0V_1+V_1^T\rho&space;_0s_0s_0^TV_1+\rho&space;_1s_1s_1^T&space;\end{align*}" title="\begin{align*} H_2 &= V_1^TH_1V_1+\rho _1s_1s_1^T\\ &= V_1^T\left ( V_0^TH_0V_0+\rho _0s_0s_0^T \right )V_1+\rho _1s_1s_1^T\\ &= V_1^TV_0^TH_0V_0V_1+V_1^T\rho _0s_0s_0^TV_1+\rho _1s_1s_1^T \end{align*}" alt="" />

若此时。仅仅保留近期的步：

\begin{align*}&space;H_{k+1}&space;&=&space;\left&space;(&space;V_k^TV_{k-1}^T\cdots&space;V_{k-m}^T&space;\right&space;)H_0\left&space;(&space;V_{k-m}\cdots&space;V_{k-1}V_k&space;\right&space;)\\&space;&+&space;\left&space;(&space;V_k^TV_{k-1}^T\cdots&space;V_{k-m}^T&space;\right&space;)\rho&space;_1s_1s_1^T\left&space;(&space;V_{k-m}\cdots&space;V_{k-1}V_k&space;\right&space;)\\&space;&+&space;\cdots&space;\\&space;&+&space;V_k^T\rho&space;_{k-1}s_{k-1}s_{k-1}^TV_k\\&space;&+&space;\rho&space;_ks_ks_k^T&space;\end{align*}" title="\begin{align*} H_{k+1} &= \left ( V_k^TV_{k-1}^T\cdots V_{k-m}^T \right )H_0\left ( V_{k-m}\cdots V_{k-1}V_k \right )\\ &+ \left ( V_k^TV_{k-1}^T\cdots V_{k-m}^T \right )\rho _1s_1s_1^T\left ( V_{k-m}\cdots V_{k-1}V_k \right )\\ &+ \cdots \\ &+ V_k^T\rho _{k-1}s_{k-1}s_{k-1}^TV_k\\ &+ \rho _ks_ks_k^T \end{align*}" alt="" />

这样在L-BFGS算法中。不再保存完整的

H_k" title="H_k" alt="" />。而是存储向量序列和。须要矩阵时，使用向量序列

\left&space;\{&space;s_k&space;\right&space;\}" title="\left \{ s_k \right \}" alt="" style="font-family:KaiTi_GB2312; font-size:18px" />和计算就能够得到。而向量序列和也不是全部都要保存，仅仅要保存最新的步向量就可以。

四、L-BFGS算法中的方向的计算方法

五、实验仿真

lbfgs.py

#coding:UTF-8
 
from numpy import *
from function import *
 
def lbfgs(fun, gfun, x0):
    result = []#保留终于的结果
    maxk = 500#最大的迭代次数
    rho = 0.55
    sigma = 0.4
 
    H0 = eye(shape(x0)[0])
 
    #s和y用于保存近期m个，这里m取6
    s = []
    y = []
    m = 6
 
    k = 1
    gk = mat(gfun(x0))#计算梯度
    dk = -H0 * gk
    while (k < maxk):
        n = 0
        mk = 0
        gk = mat(gfun(x0))#计算梯度
        while (n < 20):
            newf = fun(x0 + rho ** n * dk)
            oldf = fun(x0)
            if (newf < oldf + sigma * (rho ** n) * (gk.T * dk)[0, 0]):
                mk = n
                break
            n = n + 1
 
        #LBFGS校正
        x = x0 + rho ** mk * dk
        #print x
 
        #保留m个
        if k > m:
            s.pop(0)
            y.pop(0)
 
        #计算最新的
        sk = x - x0
        yk = gfun(x) - gk
 
        s.append(sk)
        y.append(yk)
 
        #two-loop的过程
        t = len(s)
        qk = gfun(x)
        a = []
        for i in xrange(t):
            alpha = (s[t - i - 1].T * qk) / (y[t - i - 1].T * s[t - i - 1])
            qk = qk - alpha[0, 0] * y[t - i - 1]
            a.append(alpha[0, 0])
        r = H0 * qk
 
        for i in xrange(t):
            beta = (y[i].T * r) / (y[i].T * s[i])
            r = r + s[i] * (a[t - i - 1] - beta[0, 0])
 
        if (yk.T * sk > 0):
            dk = -r            
 
        k = k + 1
        x0 = x
        result.append(fun(x0))
 
    return result

function.py

#coding:UTF-8
'''
Created on 2015年5月19日
 
@author: zhaozhiyong
'''
 
from numpy import *
 
#fun
def fun(x):
    return 100 * (x[0,0] ** 2 - x[1,0]) ** 2 + (x[0,0] - 1) ** 2
 
#gfun
def gfun(x):
    result = zeros((2, 1))
    result[0, 0] = 400 * x[0,0] * (x[0,0] ** 2 - x[1,0]) + 2 * (x[0,0] - 1)
    result[1, 0] = -200 * (x[0,0] ** 2 - x[1,0])
    return result

testLBFGS.py

#coding:UTF-8
'''
Created on 2015年6月6日
 
@author: zhaozhiyong
'''
 
from lbfgs import *
 
import matplotlib.pyplot as plt  
 
x0 = mat([[-1.2], [1]])
result = lbfgs(fun, gfun, x0)
print result
 
n = len(result)
ax = plt.figure().add_subplot(111)
x = arange(0, n, 1)
y = result
ax.plot(x,y)
 
plt.show()

实验结果

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參考文献

1. libLBFGS: a library of Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS)